Toc

There
are two components to calculus. One is the measure the rate of change at any
given point on a curve. This rate of change is called the derivative. The
simplest example of a rate of change of a function is the slope of a line. We
take this one step further to get the rate of change at a point on a line. The
other part of calculus is used to measure the exact area under a curve. This is
called the integral. If you wanted to find the area of a semicircle, you could
use integration to get the answer.

The two parts; the derivative and the integral are inverse functions of each
other. That is, they cancel each other out.

Just as (x^{2})^{1/2}=x,

the derivative of (integral (x)) = x and

derivative of (integral (f (x)) = f(x).

The
derivative is a composite function. This means it is a function acting on
another funcion. In fact, the function, is the input instead of just x. The
derivative, then takes a type of formula and turns it into another simiilar type
of formula. So, a polynomial will always yield a polynomial derivative. A
trigonomic function will always yield a trigonomic derivative. There are a few
exceptions, but this is generally the case. This is also true for the integral

Geometrically, the
derivative can be perceived as the slope of the tangent line to a curve at a
given point. This is roughly how steep the curve is at a given point. We can
easily find the rate of change of a line just by finding the slope. But, most
formulas are not as simple as a line and they're usually curved. We use the
basic formula of a line to get the derivative. If you remember the slope of a
line is:

y_{2}-y_{1}

x_{2}-x_{1}

where the change in height is divided
by the change in width. The derivative is derived by using this formula and
taking x_{2}to be infinitely close to x_{1}. When plugging
differnt formulas into this formula, they are translated to another similar type
formula. The general formula for the derivative is:

The derivative of f(x) = limit as x_{2} goes to x_{1} of the
formula:

f(x_{2})-f(x)_{1}

x_{2}-x_{1}

This can be written in shorthand as:

f'(x)
=
lim
f(x_{2}) - f(x_{1})

x_{2}
-->x_{1}
x_{2}
- x_{1}

So, what would happen if we plug in a
formula like x^{2} in the above equation? Remember, we're dealing with a
composite funcion, so f(x) is substituted instead of x. So, our derivative of
f(x) = x^{2} is:

f'(x)
=
lim
x_{2}^{2} - x_{1}^{2}

x_{2}
--> x_{1}
x_{2}
- x_{1}

**= 2x**

There is a proof to this, but I don't think that is necessary in this brief
introduction to calculus. But, did you notice what happened? (x<SUP2<
sup>)' becomes 2x. The more general formula for the derivative of a
polynomial term ,with no constant term in front of the x, is

**(x ^{n})' = nx^{n-1}**So, if f(x) = x

f'(x) = (3)(5)x

(cx

where c is any constant.There are other types of derivatives. Some of the main ones are:

If f(x) = e^{x}, then f'(x) =
e^{x}

If f(x) = sin x, then f'(x) = cos xand if f(x) = sin nx, then f'(x) = n cos nx

If f(x) = cos x, then f'(x) = -sin x andif f(x) = cos nx, then f'(x) = -nsin
xThere are many other types, but these are a few of the main ones to get you
started.